[Paper Review] The influence of the coefficients of a system of coupled wave equations with fractional damping on its stabilization

Mohammad Akil, Mouhammad Ghader+1 more|arXiv (Cornell University)|Aug 30, 2018

Stability and Controllability of Differential Equations44 references3 citations

TL;DR

This paper investigates the stabilization of a system of two one-dimensional wave equations coupled by velocities, with only one boundary fractional damping of Caputo type (order α ∈ (0,1)). Using a frequency domain approach combined with a multiplier method, it establishes optimal polynomial energy decay rates that depend critically on the coupling parameter b, the wave speed ratio a, and the fractional damping order α. The key result is that when wave speeds are equal (a=1), the decay rate is t⁻²⁄⁽¹⁻ᵃ⁾ if b ∉ πℤ, and t⁻²⁄⁽⁵⁻ᵃ⁾ if b ∈ πℤ; for unequal speeds, the decay rate is t⁻²⁄⁽⁵⁻ᵃ⁾ under various conditions on a and b.

ABSTRACT

In this work, we consider a system of two wave equations coupled by velocities in one-dimensional space, with one boundary fractional damping. First, we show that the system is strongly asymptotically stable if and only if the coupling parameter b of the two equations is outside a discrete set of exceptional real values. Next, we show that our system is not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using frequency domain approach combining with multiplier method, we prove that the energy decay rate is greatly influenced by the nature of the coupling parameter b, the arithmetic property of the ratio of the wave propagation speeds a, the order of the fractional damping. Indeed, under the equal speed propagation condition, we establish an optimal polynomial energy decay rate. Furthermore, when the wave propagate with different speeds, under some arithmetic conditions on the ratio of the wave propagation speeds, we prove that the energy of our system decays polynomially to zero.

Motivation & Objective

• To analyze the influence of coefficients—specifically the coupling parameter b, wave speed ratio a, and fractional damping order α—on the stabilization of a system of two wave equations coupled by velocities.
• To determine whether the system exhibits uniform or only polynomial energy decay, given only one boundary fractional damping.
• To establish optimal polynomial decay rates for the energy of smooth initial data under different conditions on b, a, and α.
• To investigate the role of arithmetic properties of √a (rational vs. irrational) and smallness of b in determining the decay rate.
• To extend existing results on scalar fractional damped wave equations to the more complex case of coupled systems with indirect damping.

Proposed method

• Employing a frequency domain approach to analyze the resolvent growth of the system's generator, which links spectral properties to long-time energy decay.
• Applying a multiplier method to derive energy estimates and control boundary terms, particularly involving the fractional derivative at x=1.
• Using the Riesz basis property and spectral analysis to characterize the eigenvalues and eigenfunctions of the system's operator.
• Establishing a contradiction argument via the Riemann-Lebesgue lemma and Diophantine approximation to rule out uniform decay and prove polynomial decay.
• Introducing a parameter-dependent multiplier to handle the coupling terms and control the energy evolution in the frequency domain.
• Analyzing the asymptotic behavior of the resolvent norm in the complex plane, particularly near the imaginary axis, to derive decay estimates.

Experimental results

Research questions

• RQ1 Under what conditions on the coupling parameter b is the system of coupled wave equations with fractional damping strongly asymptotically stable?
• RQ2 Why does the system fail to be uniformly (exponentially) stable, and what determines the polynomial decay rate of the energy?
• RQ3 How does the ratio of wave propagation speeds a affect the energy decay rate, especially when a ≠ 1?
• RQ4 What is the role of the arithmetic nature of √a (rational or almost all irrational) in determining the decay rate?
• RQ5 How does the size of the coupling parameter b influence the decay rate when a is rational but √a is irrational?

Key findings

• The system is strongly asymptotically stable if and only if the coupling parameter b is not in the discrete set πℤ.
• The system is not uniformly stable, and the energy decays polynomially, not exponentially.
• When wave speeds are equal (a=1), the optimal energy decay rate is t⁻²⁄⁽¹⁻ᵃ⁾ if b ∉ πℤ, and t⁻²⁄⁽⁵⁻ᵃ⁾ if b ∈ πℤ.
• When wave speeds differ (a ≠ 1), the energy decays like t⁻²⁄⁽⁵⁻ᵃ⁾ for all rational √a, and for almost all irrational √a.
• For a ∈ ℚ with √a ∉ ℚ and b sufficiently small, the decay rate remains t⁻²⁄⁽⁵⁻ᵃ⁾.
• The decay rate t⁻²⁄⁽⁵⁻ᵃ⁾ is optimal under the stated conditions, and the paper conjectures slower decay in remaining cases.